Logistic growth

Question 1

“y is directly proportional to x”
“y varies directly as x”
“y is proportional to x”

Answer 1

y = k * x^2

Question 2

“y is inversely proportional to x”
“y varies inversely as x”

Answer 2

y = k / [ x (1 - x) ]

Question 3

“z varies jointly as x and y”

Answer 3

z = k (x - y)

Question 4

Logistic growth formula (normal and derivative)

Answer 4

y = L / [ 1 + be^(-kx) ]

dy/dx = ky (1 - y/L)

Question 5

Where does the fastest growth occur?

Answer 5

Point of inflection
OR
y = L / 2

Question 6

k > 0 vs k < 0

Answer 6

k > 0 = growth
k < 0 = decay

Question 7

What does L do to a graph?

Answer 7

L is carrying capacity and raises horizontal tangent line

Question 8

What does b do to a graph?

Answer 8

Shifts the point of inflection (horizontally)

b IS ALWAYS GREATER THAN ZERO

Question 9

When dividing a proportionality constant by a constant, what does it turn into?

Answer 9

Another constant
Ex) k / 3 = F

Question 10

Shortcut to dy/dx = k * y

Answer 10

y = De^(kx)

Ex) dy/dx = 4y / 9x
dy / 4y = dx / 9x
(then you can integrate each side seperately)

Question 11

Euler’s method (“oilers”)

Answer 11

Columns: x | y | dy/dx | dy/dx * (delta x)

Add the answer of the last column to the y column of the new row
Your final answer will be in the y column of the number desired for x.

Question 12

Ex problem: Find the particular solution, y = f(x) , of this differential equation, given that f(0) = 5

dy / dx = e^(4x) / 3y

• don’t forget plus c !!

Answer 12

∫ 3y dy = ∫ e^(4x) dx
3/2 (y^2) = 1/4 * e^(4x) + c
3/2 (-5)^2 = 1/4 * e^(4*0) + c
c = 150/4 - 1/4 = 149/4
- - - - - - - - - - - - - - - - - - - - - - -
y^2 = 2/3 [ 1/4 * e^(4x) + 149/4 ]
(then take the square root)